Math  /  Algebra

Question(D) The cost. in dollars, of producing ss desks has tripled
2. Determine if each of the following relationships can be represented by a function 1 where y=1(x)y=1(x). Gives reason for your answer. a. The base of a triangle is 4 inches, the height is xx inches, and the area is yy square inches b. A set of ordered pairs where x=x= any integer and y=y= the cube root of zz c. \begin{tabular}{c|cccccc} xx & 3 & 1 & 4 & 1 & 5 & 9 \\ \hlinef(x)f(x) & 2 & 7 & 1 & 8 & 2 & 8 \end{tabular} d. {(1,5),(3,4),(5,0),(7,16),(9,25)}\{(1,5),(3,4),(5,0),(7,16),(9,25)\}

Studdy Solution

STEP 1

What is this asking? We need to figure out which of these relationships are functions, and why! Watch out! Remember, a function means each input (xx) has only *one* output (yy).
If one input has multiple outputs, it's *not* a function!

STEP 2

1. Triangle Area
2. Cube Root
3. Table of Values
4. Set of Ordered Pairs

STEP 3

The area of a triangle is given by Area=12baseheight \text{Area} = \frac{1}{2} \cdot \text{base} \cdot \text{height} .
Here, the base is **4 inches**, and the height is xx inches.

STEP 4

So, the area, which is yy, can be written as y=124x y = \frac{1}{2} \cdot 4 \cdot x .

STEP 5

Simplifying, we get y=2x y = 2 \cdot x .
For every value of xx, there's only *one* value of yy.
So, *yes*, this is a **function**!

STEP 6

We're told xx can be any integer and yy is the cube root of xx.
This means y=x3 y = \sqrt[3]{x} .

STEP 7

Every integer has only *one* cube root.
For example, the cube root of 8 is 2, and the cube root of -8 is -2.
There's no ambiguity!

STEP 8

Therefore, this relationship *is* a **function**!

STEP 9

Let's look closely at the table.
We see that when x=1x = 1, f(x)f(x) can be **7** *or* **8**.
Uh oh!

STEP 10

This means the input x=1x = 1 has *two* different outputs.
That breaks the function rule!

STEP 11

So, this relationship is *not* a **function**.

STEP 12

Remember, ordered pairs are written as (x,y)(x, y).
Let's check if any xx value appears more than once with different yy values.

STEP 13

We have (1,5)(1, 5), (3,4)(3, 4), (5,0)(5, 0), (7,16)(7, 16), and (9,25)(9, 25).
Each xx value is unique!

STEP 14

Since each input has only *one* output, this set of ordered pairs *is* a **function**!

STEP 15

a. **Function** b. **Function** c. **Not a function** d. **Function**

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